# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A prime ideal principle in commutative algebra. (English) Zbl 1168.13002

Let $R$ be a commutative ring with identity. It is a “metatheorem” in commutative algebra that an ideal maximal with respect to some property is often prime. Of course the best known and probably most important is Krull’s result that an ideal maximal with respect to missing a multiplicatively closed set is prime. Also, an ideal maximal with respect to not being principal, invertible, or finitely generated or an ideal maximal among annihilators of nonzero elements of a module is prime. This delightful paper actually gives such a metatheorem, the Prime Ideal Principle.

Let $𝔉$ be a family of ideals of $R$ with $R\in 𝔉$. Then $𝔉$ is an Oka family (resp., Ako family) if for an ideal $I$ of $R$ and $a,b\in R$, $\left(I,a\right),\left(I:a\right)\in 𝔉$ implies $I\in 𝔉$ (resp., $\left(I,a\right),\left(I,b\right)$ implies $\left(I,ab\right)\in 𝔉$). The Prime Ideal Principle states that if $𝔉$ is an Oka or Ako family, then the complement of the family ${𝔉}^{c}\subseteq \text{Spec}\left(R\right)$. Hence if $𝔉$ is an Oka or Ako family in $R$ and every nonempty chain of ideals in $𝔉$ has an upper bound in $𝔉$ and all primes belong to $𝔉$, then all ideals of $R$ belong to $𝔉$.

From these two results we recapture the results listed above plus many more and the well known consequences such as $R$ is noetherian (resp. a Dedekind domain, a PIR) if every nonzero prime ideal is finitely generated (resp., invertible, principal). The paper studies Oka and Ako families and related types of families in detail.

Many more applications of the Prime Ideal Principal are given, some of them new such as the following: a ring $R$ is Artinian if and only if for each prime ideal $P$ of $R$, $P$ is finitely generated and $R/P$ is finitely cogenerated. The work is also interpreted in terms of categories of cyclic modules.

This paper was a joy to read and should be read by all those interested in commutative algebra.

##### MSC:
 13A15 Ideals; multiplicative ideal theory
prime ideal